Residential refrigerator efficiency is a very well studied field because of the impact these appliances have on overall energy consumption, but I have not found any evidence of a specific study on this strategy.
However, some reputable organisations do support the claim. According to the Consumer Energy Center of the California Energy Commission:
A full refrigerator retains cold better than an empty one. If your refrigerator is nearly empty, store water-filled containers inside. The mass of cold items will enable the refrigerator to recover more quickly after the door has been opened. On the other hand, don't overfill it, since that will interfere with the circulation of cold air inside. The simplest solution is to buy the right size for your family in the first place.
—Consumer Energy Center – California Energy Commission
Portland General Electric agrees:
Keep it full.
Full refrigerators run more efficiently. To fill space, you can fill empty milk jugs with water and place in the refrigerator.
—Kitchen Appliances: Refrigerators | PGE
So the basic theory is that having a cold mass inside will help when the door is opened (which I find easy to believe, similarly to how the sea makes temperature milder), and that this saving will pay back the cost of making such mass cold in the first place (which I find possible, but nowhere certain).
These are the best references I could find and while they do come from reputable organisations, they don't seem to be based on experiments.